This paper proposes a novel signed $\beta$-model for directed signed network, which is frequently encountered in application domains but largely neglected in literature. The proposed signed $\beta$-model decomposes a directed signed network as the difference of two unsigned networks and embeds each node with two latent factors for in-status and out-status. The presence of negative edges leads to a non-concave log-likelihood, and a one-step estimation algorithm is developed to facilitate parameter estimation, which is efficient both theoretically and computationally. We also develop an inferential procedure for pairwise and multiple node comparisons under the signed $\beta$-model, which fills the void of lacking uncertainty quantification for node ranking. Theoretical results are established for the coverage probability of confidence interval, as well as the false discovery rate (FDR) control for multiple node comparison. The finite sample performance of the signed $\beta$-model is also examined through extensive numerical experiments on both synthetic and real-life networks.

中文翻译：

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有向符号网络中符号$β$-模型的有效估计和推理

本文提出了一种新的有向签名网络的签名$\beta$-model，该模型在应用领域中经常遇到，但在文献中大部分被忽视。所提出的有符号$\beta$-模型将有向符号网络分解为两个无符号网络的差，并在每个节点中嵌入两个潜在因子，用于状态和输出。负边缘的存在导致非凹对数似然，并且开发了一步估计算法以促进参数估计，该算法在理论上和计算上都是有效的。我们还开发了一个推理程序，用于在有符号的 $\beta$ 模型下进行成对和多节点比较，这填补了节点排名缺乏不确定性量化的空白。建立了置信区间覆盖概率的理论结果，以及用于多节点比较的错误发现率 (FDR) 控制。还通过在合成网络和真实网络上进行的大量数值实验来检验带符号的 $\beta$ 模型的有限样本性能。